An article printed in Starburst magazine from the UK has some information about the special effects from Harry Potter and the Goblet of Fire.
Creature Effects Supervisor Nick Dudman describes some of the effects in the movie:
The wings are literally 'puppeteered' from behind, and the tail? – it's a tail on a stick basically. The beast's front half is animated via a computer performance system. Other achievements by the props department include silicone aquatronic dummies of four of the youngsters – Hermione, Ron, Cho-Chang, and Fleur – for an underwater sequence that was filmed in a 60 by 20 foot tank.
Visual Effects Supervisor Jimmie Mitchell comments on the fact that while the fourth film will have no problems standing up against the other Harry Potter movies as it incorporates elements from second and third films.